Why a real lens has no focal point
One collimated bundle through an ideal lens and through a real N-BK7 singlet of the same focal length — a point against a 30 mm smear.
Open in the canvas →Background
The lensmaker's equation gives a lens one focal length, and the thin-lens construction sends every ray through one point. Both are approximations that hold only near the axis. A real lens is bounded by spheres, and a sphere is the wrong shape: it bends a ray that strikes it far from the axis too strongly, so the rim of the lens focuses closer than the centre does.
That is spherical aberration, and unlike chromatic aberration it does not go away with a single colour — it is there in monochromatic light, for a perfectly made lens, as a consequence of the shape alone. There is no plane anywhere along the axis where the light comes to a point. The best you get is the circle of least confusion, the plane where the blur is smallest.
What this setup demonstrates
Both rows start identically: a monochromatic point source at the front focus of a collimator, which turns it into a 90 mm bundle of parallel rays. Only the lens under test differs, and both have the same focal length, so any difference is the shape of the glass and nothing else.
The top row uses the idealised lens element. Every ray crosses the
axis at exactly the same place, and the screen at that plane catches a point.
The bottom row uses a real N-BK7 thicklens with the same power. The
rays now cross the axis at five distinct places spread over 31 mm: the rim
focuses 31 mm short of the paraxial focus, and the classic caustic opens up
between the two. At the paraxial plane where the screen sits, that same light is
spread over about 30 mm. Even at its tightest, 24 mm before the screen, the
spot is still about 7 mm across.
Select the singlet and shrink its aperture — the blur collapses far faster than the aperture does, because the transverse blur grows as the cube of the ray height. That is the whole reason stopping a lens down sharpens the image, and why a fast lens is so much harder to build than a slow one. Bending the lens by making the two radii unequal at constant power also helps, and splitting the power over more surfaces helps most of all — which is what the lens-group element is for.
This singlet is deliberately fast — about f/1.3 — so the caustic is obvious at a glance; a normal f/8 lens would show a blur too small to see at this scale. The tracer samples ten rays, so the caustic is drawn as a handful of distinct crossings rather than the continuous surface it really is, and the drawing carries no information about how the energy is distributed within the blur: in a real spot most of the light piles up near the circle of least confusion rather than spreading evenly. Only spherical aberration is on show here — the bundle is on-axis, so coma, astigmatism, and field curvature never appear, and the single wavelength hides chromatic aberration entirely. Diffraction is not modelled at all, so the ideal lens focuses to a mathematical point rather than to an Airy disc.